The 14th Amendment... And U.S. Code at the time of his birth...
If
one parent is a US citizen, and the other is not, and the parents are married, then the current law says the child is a US citizen if the American parent was physically present in the US for one or more periods of time totalling at least five years, at some time or times in his or her life prior to (but not necessarily
immediately prior to) the child's birth. Additionally, at least two years out the required five years of physical presence must have taken place after the parent's 14th birthday; thus, for example, a parent who was born and grew up in the US, but who left before reaching age 16 and never returned, doesn't meet the requirement.
Prior to 14 November 1986, the physical presence requirement in this case was ten years (instead of five) -- including five years (instead of two) spent after the parent's 14th birthday. The requirement was reduced in 1986, but the change did
not retroactively make US citizenship available to people born previously who did not meet the old requirement. (Congress's intent
not to make this change retroactive was affirmed in 1988 with the passage of Public Law
100-525, § 8(d), 102 Stat. 2619).
A description of the US naturalization oath is given in Section 337(a) of the INA [
8 USC § 1448(a)]. Of particular relevance to the dual citizenship issue is that, as part of the oath, a new citizen must pledge "to renounce and abjure absolutely and entirely all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which the applicant was before a subject or citizen."